Constructions (OCR GCSE Maths): Revision Notes

Example Constructing a Triangle Given Three Sides (SSS Construction)

Problem: Construct the triangle $PQR$ with sides: $PQ=18 cm$, $PR=10 cm$, and $QR=14 cm$.

Steps:

1. Draw the Base:

• Start by selecting the longest side $PQ=18$cm as the base.
• Use your ruler to draw a straight line $PQ$ that is $18$cm long. Label the ends as $P$ and $Q$.

2. Draw the First Arc:

• Set your compass to the length of $PR=10$cm.

  

• Place the pointy end of the compass on $P$ and draw an arc above the line $PQ$.

3. Draw the Second Arc:

• Without changing the compass width, set your compass to $QR=14$cm.

  

• Place the pointy end on $Q$ and draw another arc that intersects the first arc.

4. Complete the Triangle:

• Label the intersection of the arcs as $R$.
• Draw straight lines from $P$ to $R$ and from $Q$ to $R$ to form the triangle $PQR$.
• Verify the lengths with your ruler to ensure accuracy.

Example

Constructing a Perpendicular Bisector

Problem: Construct a perpendicular bisector to the line $PQ$.

Steps:

5. Set Up the Compass:

• Place the pointy end of your compass at point $P$
• Open the compass to a width that is more than half the length of line $PQ$.
• Draw an arc above and below the line $PQ$.

6. Repeat from the Other Endpoint:

• Without changing the compass width, place the pointy end at point $Q$.
• Draw another arc that crosses the first set of arcs above and below the line.

3. Draw the Perpendicular Bisector:

• Using your ruler, draw a straight line through the two points where the arcs intersect. This line is the perpendicular bisector.

Key Points:

• Ensure Accuracy: The compass width must remain the same when drawing arcs from both $P$ and $Q$.
• Label Clearly: Mark the intersection points of the arcs and label the bisector appropriately. Result: The line you have drawn is the perpendicular bisector of $PQ$ meaning it cuts $PQ$ into two equal parts and forms a right angle with it.

Example

Constructing an Angle Bisector

Problem: Construct an angle bisector for the angle made by lines $PQ$ and $PR$.

Steps:

7. Draw the First Arc:

• Place the pointy end of your compass at the angle's vertex, $P$.
• Draw an arc that crosses both lines $PQ$ and $PR$.
• This arc creates two points of intersection on $PQ$ and $PR.$

8. Draw Two More Arcs:

• Without changing the compass width, place the pointy end on one of the intersection points and draw an arc within the angle.
• Repeat the process from the second intersection point, ensuring that this arc crosses the first arc.
• The intersection of these two arcs inside the angle is crucial.

9. Draw the Angle Bisector:

• Using your ruler, draw a straight line from the vertex $P$ through the intersection of the two arcs inside the angle.

• This line is the angle bisector. Key Points:

• Consistency: The compass width must remain the same when drawing both arcs inside the angle.

• Labeling: Clearly label the intersection points and the bisector to avoid confusion. Result: The line you have drawn is the bisector of the angle $∠PQR$ meaning it divides $∠PQR$ into two equal angles.